Frequency Weighted Sub-System Model Truncation to Minimize Systems Level Model Reduction Errors

Ben Shapiro, Assistant Professor, Aerospace Engineering, University of Maryland

This subject arose out of a model reduction need in packaged electronics (cell-phones, laptops, palm-pilots, etc). We were trying to create a set of component reduced-order models that we could inter-connect in various combinations to examine systems level tradeoffs. For a simple heat equation example, we found that the sub-system errors were greatly amplified. Even if the component models were correct to 5%, the system errors would be 20% or even unstable.

The talk will present a solution to this problem for the simplest possible case. The focus is on linear sub-systems connected in a linear fashion. First we show that the sub-system model reduction errors are essentially amplified by the full system frequency response. From this there follows a stability theorem: in order to preserve entire system stability under sub-system model truncation, one must keep the norm of the sub-system model reduction error weighted by the entire system frequency response below one. Thus we propose that the sub-system model reduction problem be weighted by a (typically rough) estimate of the entire system frequency response.

The basic idea is demonstrated for a simple example consisting of a random, stable, sub-system repeated 3 times in a loop. It is shown that if the random sub-system is model reduced using unweighted balanced truncation, then the resulting error is amplified, and the model reduction error for the entire system is unacceptable. When the sub-system is model reduced by weighted balanced truncation, using a simple estimate of the entire system frequency response, then the entire system error is small. In fact, for the weighted case in our example, the system error now lies below the sub-system error.